CPC VI -- Chemical Process Control VI

Nonlinear Model Reduction for Optimization Based Control of Transient Chemical Processes 15
d 1 (t)
optimizing
feedback
control
system
optimizing
feedback
controller 1
uc,1 (t)
1 (t)
c,1
subprocess
1
co
decision
maker
g, m
,
coordinator
d 2 (t)
co
optimizing
feedback
controller 2
uc,2 (t)
2 (t)
c,2
subprocess
2
Figure 2: Horizontal decomposition, decentralized
optimization approach. δc and δco refer to the feedback
control and to the coordination sampling times.
a decentralization of the control problem typically (but
not necessarily, e.g. Lee et al. (2000)) oriented at the
functional constituents of a plant (e.g. the process units).
Coordination is required to guarantee that the optimal
value of the objective reached by a centralized optimizing
control system (see Figure 1) can also be achieved by
decentralized dynamic optimization. Various coordination
strategies for dynamic systems have been described,
for example, by Findeisen et al. (1980). Figure 2 shows
one possible structure, where the coordinator adjusts the
objective functions of the decentralized optimizing feedback
controllers to achieve the ”true” optimum of the
centralized approach.
Vertical decomposition refers to a multi-level separation
of the problem (1), (2) with respect to different
time-scales. Typically, base control, predictive reference
trajectory tracking control, and dynamic economic optimization
could be applied with widely differing sampling
rates in the range of seconds, minutes, and hours
(see Findeisen et al. (1980) for example). According to
Helbig et al. (2000b), the feasibility of a multiple timescale
decomposition does not only depend on the dynamic
properties of the autonomous system but also on
the nature of the exogenous inputs and disturbances. If,
for example in a stationary situation, the disturbance
can be decomposed into at least two contributions,
d(t) = d0(t) + ∆d(t) , (3)
a slow trend d0(t) fully determined by slow frequency
contributions and an additional zero mean contribution
optimizing feedback
control system
time scale
separator
0 (t)
long time
scale model
update
d 0 (t)
(t)
short time
scale model
update
d(t)
c
(t)
decision
maker
g, m ,
optimal
trajectory
design
gc, x d (t) y d (t), u d (t)
tracking
controller
uc(t)
process
including
base control
Figure 3: Vertical, two time-scale decomposition of
optimization based operations support for transient
processes. δc refers to the sampling time of the tracking
controller, Ψ refers to a process performance indicator.
∆d(t) containing high frequencies, some sort of decomposition
should be feasible. Figure 3 shows a possible
structure of the optimization based operations support
system in this case. The upper level is responsible for the
design of a desired optimal trajectory xd(t), ud(t), y d(t)
whereas the lower level is tracking the trajectory set
by the upper level. Due to the time varying nature of
the disturbances d(t), feedback is not only necessary to
adjust the action of the tracking controller but also to
adjust the optimal trajectory design to compensate for
variations in d0(t) and ∆d(t), respectively. The control
action uc(t) is the sum of the desired control trajectory
ud(t) and the tracking controller output ∆u(t). Reconciliation
is based on the slow and fast contributions
η 0(t) and ∆η(t) separated by a time-scale separation
module. Control and trajectory design are typically executed
on two distinct sampling intervals δc and kδc with
integer k > 1. The performance of the controller, coded
in some indicator Ψ, needs to be monitored and communicated
to the trajectory design level to trigger an
update of the optimal trajectory in case the controller
is not able to achieve acceptable performance. Though
this decomposition scheme is largely related to so-called
composite control in the singular perturbation literature
Kokotovic et al. (1986), the achievable performance will
be determined by the way the time-scale separator is implemented.
Model Requirements
Optimization based control requires appropriate models
to implement solutions to the reconciliation and control
problems.
The model in (2) must predict the cost function, out-
d(t)